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Psychological Test and Assessment Modeling, Volume 58, 2016 (4), 567-592
The detection of heteroscedasticity in
regression models for psychological data
Andreas G. Klein1, Carla Gerhard2, Rebecca D. Büchner2,
Stefan Diestel3& Karin Schermelleh-Engel2
Abstract
One assumption of multiple regression analysis is homoscedasticity of errors. Heteroscedasticity,
as often found in psychological or behavioral data, may result from misspecification due to
overlooked nonlinear predictor terms or to unobserved predictors not included in the model.
Although methods exist to test for heteroscedasticity, they require a parametric model for specifying
the structure of heteroscedasticity. The aim of this article is to propose a simple measure of
heteroscedasticity, which does not need a parametric model and is able to detect omitted nonlinear
terms. This measure utilizes the dispersion of the squared regression residuals. Simulation studies
show that the measure performs satisfactorily with regard to Type I error rates and power when
sample size and effect size are large enough. It outperforms the Breusch-Pagan test when a
nonlinear term is omitted in the analysis model. We also demonstrate the performance of the
measure using a data set from industrial psychology.
Keywords: Heteroscedasticity, Monte Carlo study, regression, interaction effect, quadratic
effect
1
Correspondence concerning this article should be addressed to: Andreas G. Klein, Department of Psychol-
ogy, Goethe University Frankfurt, Theodor-W.-Adorno-Platz 6, 60629 Frankfurt; email: klein@psych.uni-
frankfurt.de
2Goethe University Frankfurt
3
International School of Management Dortmund Leipniz-Research Centre for Working Environment and
Human Factors
568 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel
Introduction
One of the standard assumptions underlying a linear model is that the errors are inde-
pendently identically distributed (i.i.d.). In particular, when the errors are i.i.d., they
are homoscedastic. If the errors are not i.i.d. and assumed to have distributions with
different variances, the errors are said to be heteroscedastic. A linear heteroscedastic
model is defined by:
yi=β0+β1x1i+... +βmxmi +εi,i=1,...,n,(1)
where the εiare realizations (sampled values) of error variables εthat follow a mixture
distribution with normal mixing components:
ε∼SZ.(2)
For the random variables S, Z we assume that
S>0,Z∼N(0,1),S⊥Z.(3)
We are making the following regularity assumptions for
S2
, the random variable that
models the variances of the errors:
0<E(S2)<∞,0≤var(S2)<∞.(4)
Heteroscedasticity is given when
var(S2)
takes a positive value. In contrast, homoscedas-
ticity holds if
var(S2)=0
. This definition of heteroscedasticity covers both models
with a discrete and with a continuous distribution of the variances of the errors. For the
random variable
S2
two types of heteroscedasticity can generally be distinguished: First,
there could be a specific, parametric form of heteroscedasticity where
S2
is a function of
the given predictors, such as
S2=exp(a0+a1x1+... +amxm)
. Models with this type
of parametric heteroscedasticity have been investigated in the past (cf. White, 1980).
Second, there could be an unspecific form of heteroscedasticity, where
S2
is entirely
unrelated to the observed explanatory variables. When a linear model for a specific set of
predictors is selected, heteroscedasticity of the errors may be due to different causes. For
instance, in social sciences and especially in psychological research one often deals with
learning mechanisms among individuals during the process of data collection. These
learning mechanisms are one possible source of heteroscedastic errors, because predic-
tion may be more accurate for subjects whose predictor scores are observed at a later
stage in their development. Furthermore, reasons for heteroscedasticity could be omitted
variables, outliers in the data, or an incorrectly specified model equation, for example
omitted product terms. In psychological contexts product terms in regression are often
related to overlooked or yet unidentified moderator variables.
Test for heteroscedasticity in regression models 569
In the following, we will focus on the relationship between heteroscedastic errors,
model misspecification and the distribution of the regression residuals when a (possibly
misspecified) regression model is fit to the data. In particular, we are interested in
misspecifications resulting from omitted nonlinear terms. In psychological research, for
instance, the issue of omitted interaction terms is often of particular interest, and some
methodological research has been concerned with the development of efficient estimation
methods in the past (cf. Dijkstra & Schermelleh-Engel, 2014; Klein & Moosbrugger,
2000; Klein & Muthén, 2007). Figure 1 gives an illustration of residuals resulting from
an omitted interaction term. In the upper panel of Figure 1, an interaction model has been
analyzed with a correctly specified model. In the lower panel of Figure 1, the interaction
model has been analyzed with a (misspecified) linear model, where the interaction term
was omitted. As can be seen, the dispersion of the residuals of the misspecified model is
no longer homoscedastic.
If the model in Equation
(1)
is correct and if heteroscedasticity of
ε
holds, the regular
OLS estimator for the regression coefficients still yields unbiased and consistent param-
eter estimates. However, the estimator is not efficient anymore, because the standard
errors are biased and therefore the usual inference methods are no longer accurate (cf.
Greene, 2012; White, 1980). For heteroscedastic models with correctly specified regres-
sion equation and correctly specified parametric heteroscedasticity, robust estimation
methods have been developed to deal with heteroscedastic errors. Two alternatives
exist that use either a weighted least squares estimator or a heteroscedasticity-consistent
covariance matrix (MacKinnon & White, 1985; White, 1980). In regression analysis
the distribution of the residuals depends on the heteroscedasticity of the errors and the
selection of predictors to model the data. Visible heteroscedasticity may therefore often
be a result of a misspecified regression model. In this case a modification of the model
structure might sometimes be more useful than using a robust estimation method.
There are different ways to test for heteroscedasticity in linear regression models. One
group of tests can be classified as ’model-based heteroscedasticity tests’ (cf. Greene,
2003). These tests are using a specific parametric model to specify the heteroscedasticity.
If this specification is incorrect, the tests may fail to identify heteroscedasticity. Three
well-known statistical tests exist that are used for such parametric models: the Wald test,
the likelihood ratio test, and the Lagrange multiplier test (cf. Engle, 1984; Greene, 2003;
Wald, 1943). Another group of tests, which is able to detect heteroscedasticity in a more
general form, can be called ’residual-based heteroscedasticity tests’ (cf. Greene, 2003).
Most of these tests are only available for categorical predictors (cf. Rosopa, Schaffer,
& Schroeder, 2013) and are not suitable for our purposes. For both categorical and
continuous predictors, two tests remain; the Breusch-Pagan and the White test. These
tests can be represented by an auxiliary regression equation that uses some function
of the estimated residuals as dependent variable and various functions of the proposed
570 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel
explanatory variables as predictor variables
1
. The well-known Breusch-Pagan test was
proposed by Breusch and Pagan (1979) and by Cook and Weisberg (1983). It has been
developed independently in the econometrics and statistics literature (cf. Rosopa et al.,
2013). The Breusch-Pagan test tests the null hypothesis that the residuals’ variances
are unrelated to a set of explanatory variables versus the alternative hypothesis that
the residuals’ variances are a parametric function of the predictor variables. The test
can be represented in an auxiliary regression form, in which the squared residuals of
the proposed model are regressed on the predictors believed to be the cause of the het-
eroscedasticity. The common White test has been proposed by White (1980), where the
squared OLS-residuals are regressed on all distinct predictors, cross products, squares
of predictors, and the intercept. The test statistic is given by the coefficient of determi-
nation of the auxiliary regression multiplied by the sample size (
nR2
). The statistic of
the White test is chi-square distributed with degrees of freedom equal to the number
of predictors in the auxiliary regression. However, these common heteroscedasticity
tests do not solve the problem of detecting heteroscedasticity that is caused by omitted
predictors. Therefore, Klein and Schermelleh-Engel (2010) proposed the
Zhet
statistic in
the context of structural equation modeling. This statistic is potentially suitable to detect
heteroscedasticity caused by omitted predictors in structural equation models. However,
in some preliminary studies
Zhet
showed an undesirably low power in the detection of
heteroscedasticity.
The objective of the current paper is to fill that gap of detecting unsystematic het-
eroscedasticity that relates to omitted predictors or yet unanalyzed nonlinear relation-
ships and explore a basic measure of heteroscedasticity. It does not require a specification
of the heteroscedasticity. Instead, the measure makes direct use of the dispersion of the
squared residuals and an additional auxiliary regression is not necessary. In addition, a
second goal for this paper is to find an approach that can be applied easily.
1
Please note that tests for heteroscedasticity presented in original literature with asymptotic chi-square
distributions, such as likelihood ratio, Wald or Lagrange multiplier test, are asymptotically equivalent to
the auxiliary regression approach (cf. Engle, 1984).
Test for heteroscedasticity in regression models 571
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−2−101234
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Residuals
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−2−101234
−3−2−10 1 2 3
y
^
Residuals
Figure 1: Scatter plot of the residuals with modeled interaction term (top) and with an
omitted interaction term (bottom). Data generated for population model
y=0.5+0.5x1+0.3x2+0.4x1x2+e, with n= 400 and e∼N(0,0.16).
572 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel
Heteroscedasticity measure
In this section, we introduce the measure
hhet
to test for heteroscedasticity of the errors.
The measure
hhet
is intended to measure a possible deviation from homoscedasticity. If
the errors are heteroscedastic, they have distributions with different standard deviations,
and one may then expect that the variance of the squared regression residuals
e
tends to
be greater than it does when the residuals are homoscedastic. After conducting an OLS
regression, the OLS residuals
ei(i=1,...,n)
are available for all
n
cases, and we have
¯e=0. We consider
var(e2)
var(e)2≈n−1Σ(e2
i−e2)2
(n−1Σ(ei−e)2)2(5)
=n−1Σe4
i−(n−1Σe2
i)2
(n−1Σe2
i)2
=n−1Σe4
i
(n−1Σe2
i)2−1
=ˆ
γ−1,
where
ˆ
γ
is the common sample-based fourth standardized moment of the residuals.
ˆ
γ
is
an estimator for the kurtosis of
e
(cf. Davidson & MacKinnon, 1993). Originally, the
estimator
ˆ
γ
has been formulated for independent
ei
values, where it is shown that
ˆ
γ
is
asymptotically normally distributed (cf. Davidson & MacKinnon, 1993). Here, because
the OLS residuals meet the constraint
e=0
, they are not independently distributed. How-
ever, we confirmed in simulation studies the asymptotic behavior of
ˆ
γ
when calculated
from OLS residuals for sufficiently large sample sizes (
n≥100
). Therefore, we can
adopt the known asymptotic distribution
ˆ
γ∼N(3,24/n)(6)
(cf. Davidson & MacKinnon, 1993) for
ˆ
γ
based on the OLS residuals. We define the
measure hhet as
hhet :=n
24 (ˆ
γ−3),(7)
so that
hhet ∼N(0,1).(8)
In case of heteroscedastic residuals, it can be shown that, asymptotically,
hhet
takes a
value greater than zero. To see this, it is sufficient to show that
lim
n→∞ˆ
γ
is greater than
Test for heteroscedasticity in regression models 573
three:
lim
n→∞ˆ
γ=lim
n→∞
n−1Σe4
i
(n−1Σe2
i)2(9)
=E(ε4)
(E(ε2))2
=E(S4)E(Z4)
(E(S2)E(Z2))2
=3E(S4)
(E(S2))2
=3var(S2)+(E(S2))2
(E(S2))2
=31+var(S2)
(E(S2))2>3.
Based on this result, it is indicated to use a one-tailed test for hhet .
The test we propose here does not make a specific assumption about what caused a
possible heteroscedasticity, and it does not need a specific parametric model of the
structure of heteroscedasticity. In contrast to residual-based heteroscedasticity tests,
hhet
is able to detect heteroscedasticity of the residuals that could be due to unobserved
nonlinear predictor terms.
Simulation study
A Monte Carlo study was conducted with the aim of investigating the sensitivity of
the measure
hhet
to respond to heteroscedasticity relating to omitted nonlinear terms.
The study investigates the influence of nonlinear effect size and sample size on the
performance of
hhet
. Various linear and nonlinear population models were selected for
data generation and the residuals were afterwards analyzed with
hhet
. For reasons of
comparability, the residuals of a model with an omitted unobserved quadratic predictor
were additionally analyzed with the Breusch-Pagan test. In the following, we first
introduce the population and analysis models as well as the particular design of the
study; second, we present the results about the performance of hhet .
574 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel
Population models for heteroscedasticity related to observed predictors
Different population models were used for data generation. Four population models
were chosen for estimating the sensitivity of
hhet
to respond to omitted nonlinear terms.
The first population model
MLQI
was a full nonlinear model with two linear (L), two
quadratic (Q) and one interaction term (I):
y=β0+β1x1+β2x2+β3x2
1+β4x2
2+β5x1x2+e,(10)
where
β0=.50,β1=.50
, and
β2=.30
were held constant across all simulation con-
ditions. For the variables
x1
,
x2
, and enormally distributed data were generated. The
correlation between
x1
and
x2
wasfixedto
r12 =.20
in all simulation conditions. The
variances of
x1
and
x2
were set to 1.00; the variance of ewas fixed to .40 in all condi-
tions.
MLQI
included three nonlinear terms, the effects of these terms were varied in
size correspondingly. For the first condition the effect sizes were set to
β3=β4=.10
,
and
β5=.15
; for the second condition to
β3=β4=.15
, and
β5=.20
. Combined, the
nonlinear terms explained between 10 % and 19 % of the variance in y.
The second population model
MLQ
was a nonlinear model with two linear (L) and one
quadratic effect (Q).
MLQ
is the same as
MLQI
, except for seting
β4=β5=0
. The size
of
β3
was set to .20 and .30 in two effect size conditions. The quadratic effect explains
between 9 % and 19 % of the variance in y.
The third population model
MLI
was a nonlinear model with two linear (L) and one
interaction effect (I).
MLI
is the same as
MLQI
, except for setting
β3=β4=0
. The
size of
β5
was set to .30 and .40 in two effect size conditions, this equals an explained
variance of 10 % to 18 % in y. In addition, to show the practical use of
hhet
for
greater regression coefficients,
MLI
was generated with another set of parameters. In the
additional condition the regression coefficients were set to
β0=2
,
β1=2
,
β2=1.2
, and
to
β5=1.2
with the same variances and covariances as before for the error term
e
and
the predictors x1and x2. The nonlinear term explained 18 % of the variance in y.
The fourth population model
ML
was a linear model with two linear (L) effects.
ML
is
the same as MLQI , but it included no nonlinear terms after setting β3=β4=β5=0.
Population models for heteroscedasticity unrelated to the observed predictors
In addition to investigating the performance of
hhet
to detect omitted nonlinear terms
which are related to the observed predictors, we examined the sensitivity of
hhet
to re-
spond to heteroscedastic residuals due to unobserved predictors. For the investigation of
the sensitivity of
hhet
to detect nonlinear terms of unobserved predictors, three population
Test for heteroscedasticity in regression models 575
models were chosen: First, a quadratic model
MLQ
with two linear (L) and one quadratic
effect (Q) was used as nonlinear population model:
y=β0+β1x1+β2x2+β4x2
2+e,(11)
where
β0=.50,β1=.50
, and
β2=.30
. The variables
x1,x2
, and ewere normally
distributed; the correlation between
x1
and
x2
was set to
r12 =.20
. The size of
β4
was
set to .15 and .25 in two effect size conditions.
Second, the population model
MLI
with two linear (L) and one interaction effect (I)was
used:
y=β0+β1x1+β2x2+β5x1x2+e.(12)
MLI
is the same as
MLQ
, except for setting
β3=0
, and
β5=.25
or
β5=.35
in two effect
size conditions.
Third, a linear population model
MsL
containing only a single linear predictor (sL)was
chosen:
y=β0+β1x1+e.(13)
MsL
is similar to
MLQ
, resulting from setting
β2=β4=0
, such that
MsL
included only a
single linear predictor and no nonlinear terms.
Design
The data for the population models were generated with the Rsoftware and analyzed
with the OLS estimator in Rversion 3.2.2 (R Core Team, 2015). For each condition R=
10,000 data sets were generated. Across all conditions, except the additional condition
for
MLI
, the sample size nwas 100, 200, 400, 800 or 1,200. For the additional condition
n=400
was selected. For heteroscedasticity related to the observed predictors, four
population models (
MLQI
,
MLQ
,
MLI
,and
ML
), two effects size conditions, and five
sample size conditions were implemented, and each population model was analyzed as
a correctly specified and as a misspecified model. For estimating
hhet
for models that
contain heteroscedasticity related to unobserved predictors, three population models
(
MLQ
,
MLI
, and
MsL
), two effect size conditions, and five sample size conditions were
implemented. For a power analysis the proportion of data sets was examined where
hhet
had values greater than the critical value at the 5 % level of a one-sized test (
z=1.65
).
Linear population models were analyzed by correctly specified models and by various
overparameterized nonlinear models to study the Type I error rate. The data generated
for nonlinear population models were analyzed by misspecified linear models for power
analysis and by correctly specified models for Type I error analysis. As the ordinary
residuals are scale dependent, some researchers recommend the use of internally or
576 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel
externally studentized residuals (Cook & Weisberg, 1982; Stevens, 1984). For the
simulation conditions presented here, the results for ordinary residuals and for internally
studentized residuals were very similar but are not reported in this article.
In the following we will provide the results for the measure
hhet
. Additionally, some
results for the Breusch-Pagan test and the AIC are compared with
hhet
. The formula for
the auxiliary regression in the Breusch-Pagan test is
e2
ˆ
σ2=α0+α1x1+α2x2
1+ε,(14)
where ˆ
σ2=∑e2
i
nand εis normally distributed with zero mean.
Results
In this section, we present the results of the simulation study. In addition to mean
hhet
-values, we report the Type I error rates and the power of
hhet
to detect omitted
nonlinear terms that resulted in heteroscedasticity. The power exceeded 80 % under
several conditions. The 95 % confidence interval (CI) for the error rate of a test with 5 %
nominal Type I error, for a sample of 10,000 cases, is calculated as [4.57,5.43]. For
hhet
,
the error rate turned out to be slightly inflated, because under some conditions the error
rate was lying slightly above this range.
Heteroscedasticity related to observed predictors
The following results refer to the investigation of the influence of varying nonlinear
effect size and sample size on the detection of heteroscedasticity with hhet.
Linear population model. For the linear population model
ML
the mean
hhet
-values and
Type I error rates for the different linear and nonlinear analysis models
ML,MLQ,MLI
,
and
MLQI
are listed in Table 1. The results indicate appropriate Type I error rates close
to the nominal 5 % level. Only one value was too small, and two values were slightly
too high. On average the hhet -values tended to be slightly negative.
The probability density functions presented in Figure 2 illustrate the convergence of the
distribution of
hhet
towards the standard normal distribution. For the plot, Epanechnikov
kernel functions were produced. The Epanechnikov kernel was used, because it displays
deviations from normality more clearly than the Gaussian kernel. The functions are
shown for the linear population model
ML
correctly analyzed as
ML
for sample sizes
Test for heteroscedasticity in regression models 577
Table 1:
Mean
hhet
-Values and Type I Error Rates (in Percent) as a Function of Sample Size (n) for the Linear Population
Model ML.
population
model:
ML
y=β0+β1x1+β2x2
analysis
model:
ML
y=β0+β1x1+β2x2
MLQ
y=β0+β1x1+β2x2
+β3x2
1
MLI
y=β0+β1x1+β2x2
+β5x1x2
MLQI
y=β0+β1x1+β2x2
+β3x2
1+β4x2
2+
β5x1x2
nMean Type I
error
Mean Type I
error
Mean Type I
error
Mean Type I
error
100 -0.15 3.84 -0.11 4.75 -0.10 4.82 -0.09 4.97
200 -0.09 4.60 -0.08 5.33 -0.09 5.25 -0.08 5.11
400 -0.06 5.00 -0.05 5.62 -0.05 5.57 -0.06 5.10
800 -0.06 5.16 -0.04 5.07 -0.03 5.23 -0.05 5.52
1,200 -0.04 5.20 -0.05 5.18 -0.04 5.20 -0.04 5.39
578 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel
−3−2−10123
0.0 0.1 0.2 0.3 0.4 0.5
hhet
Probability Density
n = 100
n = 200
n = 800
n = 1200
N(0,1)
Figure 2:
Estimated probability density function of
hhet
for the linear Population Model
ML
correctly analyzed as
ML
. The density functions were estimated using an
Epanechnikov kernel function.
100,200,800
and
1,200
. The density function for
n=1,200
is close to the
N(0,1)
-
density and has a kurtosis of 0.36 and a skewness of 0.42. The kurtosis for
n=100
is
3.20, whereas the skewness is 1.22. Both kurtosis and skewness decrease with greater
sample size. In the critical part of the distribution, the right hand tail, there were only
small deviations from the standard normal curve. We note that for small samples the
density curve for hhet >1.645 is slightly above or below the ideal normal density. As a
consequence, the Type I error does not deviate much from .05 (see Table 1).
Nonlinear population model. In Table 2 the results for the quadratic population model
MLQ
are presented. In addition, the power of
hhet
is listed, where
hhet
correctly indicates
the presence of heteroscedasticity in the residuals of the linear analysis model
ML
.It
appears that the Type I error rates were close to the nominal 5 % level for all sample
sizes, where one value was too high. A desirable power of 80 % was exceeded at sample
size
n=600
when the nonlinear effect size was
β3=.30
. For a quadratic effect size of
β3=.20
a power of 80 % was not reached for the listed values. Additional simulations
indicate a required sample size of n≥2,700 (not listed in Table 2).
The results for the population model
MLI
can be seen from Table 3. The Type I error
rates were again close to their nominal 5 % levels in all conditions, four values were just
outside the 95 % CI bounds. A power close to 80 % was reached for sample size of 1,200
and interaction effect size of
β5=.40
. For a small effect size (
β5=.30
) a sample size
Test for heteroscedasticity in regression models 579
Table 2:
Mean
hhet
-Values, Type I Error Rates (in Percent), and Power (in Percent) as a Function of Sample Size (n) and
Quadratic Effect Size for the Quadratic Population Model MLQ.
population
model:
MLQ
y=β0+β1x1+β2x2+β3x2
1
analysis
model:
MLQ
y=β0+β1x1+β2x2+β3x2
1
ML
y=β0+β1x1+β2x2
nonlinear
pop.
parameter:
β3=.20 β3=.30 β3=.20 β3=.30
nMean Type I
error
Mean Type I
error
Mean Power Mean Power
100 -0.12 4.59 -0.12 4.76 0.20 10.02 1.07 26.17
200 -0.10 4.87 -0.08 5.31 0.62 18.47 2.26 45.75
400 -0.07 5.09 -0.06 5.40 1.07 27.35 3.87 70.77
800 -0.05 5.29 -0.04 5.09 1.69 41.52 5.98 91.07
1,200 -0.03 5.47 -0.05 5.43 2.19 54.23 7.53 97.41
580 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel
of at least
n=4,000
is needed (not listed in Table 3). For the population model
MLI
, the
AIC
was in all cases smaller than for model
ML
and therefore confirmed the improvement
of the results compared to model
MLI
. Additionally,
hhet
was calculated for a model with
larger parameter values, i.e.,
β0=2
,
β1=2
,
β2=1.2
,
β5=1.2
and
n=400
. The power
was high (99.99 %), and Type I error rate was only slightly increased (5.62 %).
Table 4 presents the results for the full nonlinear population model
MLQI
. Type I error
rates were again close to the nominal5%level,whereas three values were slightly too
high. A power of 80 % was exceeded in samples with
n>1,000
when the nonlinear
effect sizes were
β3=β4=.15
,
β5=.20
. Models with small effects required sample
sizes of n=2,200 in order to reach a power of 80 % (not listed in Table 4).
Heteroscedasticity unrelated to observed predictors
The following results relate to the investigation of heteroscedasticity due to unobserved
predictors. The influence of nonlinear effect size and sample size on the
hhet
-values were
examined. The results of the quadratic population model were compared to results of an
analysis with the Breusch-Pagan test.
Linear population model. For the population model
MsL
with a single linear effect the
mean values of
hhet
and the Type I error rates are given in Table 5. The Type I error rates
were close to the nominal5%levelinallconditions ranging from 4.30 % to 5.62 %.
Four Type I error rates were lying outside the 95 % CI bounds.
Nonlinear population models. The power of
hhet
to detect heteroscedasticity related to an
unobserved predictor in a moderator model is provided in Table 6.
MLI
was analyzed as
MsL
, a model with a single linear predictor
x1
. Even though the second linear predictor
x2
was not included in the analysis model,
hhet
responds to the heteroscedasticity caused
by the interaction of the observed predictor
x1
and of the unobserved predictor
x2
. The
power was close to 80 % for n=800 and β3=.35.
Table 7 reports the influence of an unobserved predictor in a quadratic population model
on
hhet
. The predictor
x2
that was part of
MLQ
was not included in the analysis model
MsL
(a linear model with only a single linear predictor). The measure
hhet
responds to the
heteroscedasticity associated to omitted predictors in
MsL
. The power ranged from 16.31
%(
n=100
,
β3=.15
) to 98.59 % (
n=1,200
,
β3=.25
) and approximately exceeded
80 % for
n=500
and
β3=.25
(not listed in Table 7). For reasons of comparability, the
results for an analysis with the Breusch-Pagan test are listed too. The power values of
the Breusch-Pagan test were lower and ranged from 7.98 % to 54.62 %. The reason why
the Breusch-Pagan test does indeed have a power clearly above 5 % lies in the fact that
x1and x2were correlated in model MLQ for the simulated data.
Test for heteroscedasticity in regression models 581
Table 3:
Mean
hhet
-Values, Type I Error Rates (in Percent), and Power (in Percent) as a Function of Sample Size (n) and
Interaction Effect Size for the Population Model MLI.
population
model:
MLI
y=β0+β1x1+β2x2+β5x1x2
analysis
model:
MLI
y=β0+β1x1+β2x2+β5x1x2
ML
y=β0+β1x1+β2x2
nonlinear
pop.
parameter:
β5=.30 β5=.40 β5=.30 β5=.40
nMean Type I
error
Mean Type I
error
Mean Power Mean Power
100 -0.11 4.48 -0.12 4.40 0.06 7.85 0.41 13.62
200 -0.07 5.40 -0.09 5.49 0.33 12.37 0.93 25.09
400 -0.06 5.27 -0.05 5.49 0.59 17.37 1.61 40.18
800 -0.07 4.95 -0.04 5.76 1.04 28.15 2.57 63.45
1,200 -0.05 5.10 -0.04 5.41 1.32 36.24 3.23 77.22
582 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel
Table 4:
Mean
hhet
-Values, Type I Error Rates (in Percent), and Power (in Percent) as a Function of Sample Size (n) and
Nonlinear Effect Size for the Full Nonlinear Population Model MLQI.
population
model:
MLQI
y=β0+β1x1+β2x2+β3x2
1+β4x2
2+β5x1x2
analysis
model:
MLQI
y=β0+β1x1+β2x2+β3x2
1+β4x2
2+β5x1x2
ML
y=β0+β1x1+β2x2
nonlinear
pop.
parameter:
β3=β4=.10
β5=.15
β3=β4=.15
β5=.20
β3=β4=.10
β5=.15
β3=β4=.15
β5=.20
nMean Type I
error
Mean Type I
error
Mean Power Mean Power
100 -0.09 4.96 -0.10 4.72 0.08 8.00 0.56 17.04
200 -0.07 4.77 -0.07 5.47 0.35 13.26 1.26 29.43
400 -0.06 5.41 -0.07 5.38 0.67 19.21 2.21 48.56
800 -0.06 5.21 -0.04 5.39 1.05 27.89 3.48 72.28
1,200 -0.05 5.44 -0.04 5.48 1.37 36.22 4.39 84.81
Test for heteroscedasticity in regression models 583
Table 5:
Mean
hhet
-Values and Type I Error Rates (in Percent) as a Function of Sample
Size (n) for the Population Model MsL with a Single Linear Effect.
population
model:
MsL
y=β0+β1x1
analysis
model:
MsL
y=β0+β1x1
MLQ
y=β0+β1x1+β2x2
+β4x2
2
MLI
y=β0+β1x1+β2x2
β5x1x2
nMean Type I
error
Mean Type I
error
Mean Type I
error
100 -0.13 4.30 -0.11 4.69 -0.12 4.75
200 -0.08 5.36 -0.08 5.36 -0.10 5.12
400 -0.07 5.36 -0.06 5.17 -0.04 5.62
800 -0.04 5.23 -0.04 5.57 -0.05 4.98
1,200 -0.03 5.45 -0.04 5.33 -0.03 5.13
Table 6:
Mean
hhet
-Values and Power (in Percent) as a Function of Sample Size (n) and
Interaction Effect Size for the Population Model MLI.
population
model:
MLI
y=β0+β1x1+β2x2+β5x1x2
analysis model: MsL
y=β0+β1x1
nonlinear pop.
parameter:
β5=.25 β5=.35
nMean Power Mean Power
100 0.32 12.90 0.68 19.90
200 0.63 18.60 1.52 36.10
400 1.13 31.80 2.14 52.00
800 1.64 43.10 3.24 76.20
1,200 1.92 53.00 3.92 87.30
Under the same conditions, except for using uncorrelated predictor terms, the power of
the Breusch-Pagan test dropped below 14 %, while the power of
hhet
was unaffected (not
listed in Table 7). A comparison with the White test (not reported here) revealed that the
White test had slightly lower power than the Breusch-Pagan test. Therefore, only results
for the Breusch-Pagan test are reported here. The power of the White test ranged from
5.85 % to 36.5 %.
584 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel
Table 7: Mean hhet - and Breusch-Pagan-Values and Power (in Percent) as a Function of Sample Size (n) and Quadratic
Effect Size for the Population Model MLQ.
heterosce-
dasticity
analysis:
hhet Breusch-Pagan test1
population
model:
MLQ
y=β0+β1x1+β2x2+β4x2
2
analysis
model:
MsL
y=β0+β1x1
nonlinear
pop.
parameter:
β4=.15 β4=.25 β4=.15 β4=.25
nMean Power Mean Power Mean Power Mean Power
100 0.55 16.31 1.58 33.23 0.46 7.98 0.42 12.58
200 0.96 24.98 2.73 52.58 0.44 10.61 0.36 19.08
400 1.49 36.57 4.31 75.95 0.39 14.18 0.30 26.71
800 2.24 54.84 6.41 93.96 0.31 22.84 0.21 42.62
1,200 2.82 69.15 7.96 98.59 0.27 30.63 0.15 54.62
1The formula for the Breusch-Pagan test was e2
˜
σ2=α0+α1x1+α2x2
1+ε, where ˜
σ2=∑e2
i
nand εis normally distributed with zero mean.
Test for heteroscedasticity in regression models 585
Empirical example
To illustrate the applicability of
hhet
an empirical example is presented where the in-
fluence of job characteristics on burnout was examined. The dependent variable is
emotional exhaustion, which is considered to be one central symptom of the burnout
syndrome (Maslach & Jackson, 1981; Maslach & Leiter, 1997). Exhaustion refers to
feelings of being overextended and drained by job demands. Three predictors were
considered: Job control, work pressure, and concentration requirements. Work pressure
involves perceived time pressure and work volume, and concentration requirements refer
to employee’s experienced degree of task complexity and demands on concentration.
Participants and procedure. The study was carried out in a large civil service organization
of a federal state in Germany (Diestel & Schmidt, 2009; Schmidt & Neubach, 2009).
Participants of the study were tax collectors, recruited from a large tax and revenue
office. During work hours, questionnaires were administered to 641 employees in small
groups of about 15 people. A final sample of 461 employees provided sufficient data.
Mean age was 40.88 (SD = 10.05), 58 % of the employees were female and 89.6 % were
employed on a full-time basis.
Measures. The burnout dimension of emotional exhaustion was measured by Büssing
and Perrar’s (1992) German translation of the Maslach Burnout Inventory (Maslach,
Jackson, & Leiter, 1986). Nine items measured emotional exhaustion (e.g., ’I feel
emotionally drained from my work’). Job control was measured by five items, which
refer to the perceived extent to which an employee can choose different strategies
and methods (Jackson, Wall, Martin, & Davids, 1993; Schmidt, 2004) (e.g., ’To what
extent can you decide how to go about getting your job done?’). Work pressure and
concentration requirements, two dimensions of work load, were measured by subscales
of the Kurzfragebogen zur Arbeitsanalyse (KFZA; Short Questionnaire for Job Analysis)
instrument developed by Prümper, Hartmannsgruber, and Frese (1995). Both scales,
originally measured by two items each, were extended by constructing two additional
items for work pressure and three additional items for concentration requirements
(Schmidt & Neubach, 2009).
Results. Two regression analyses were conducted. First, a linear model was analyzed,
where ’emotional exhaustion’
(EE)
was regressed on ’work pressure’
(WP)
, ’concentra-
tion requirements’ (CR), and ’job control’ (JC):
EE =β0+β1WP+β2CR +β3JC +e(15)
The OLS regression for the linear model yielded
R2=.47
, the standardized regression
equation is:
ˆzEE =.27zWP +.32zCR −.26zJC.(16)
586 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel
All three linear effects were significant (with
t=5.93,SE =.045,p<.01
for predictor
WP
;
t=−6.95,SE =0.038,p<.01
for predictor
JC
;
t=7.13,SE =.045,p < .01
for predictor
CR
). The analysis of the residuals resulted in
hhet =1.84
for the linear
model. For
α=5%
the critical
hhet
value is
1.65
. Thus, the residuals showed significant
heteroscedasticity in the linear regression model. It can be inferred that possible modera-
tor and nonlinear effects may have been omitted in the linear model. The AIC for this
model was 1019.82.
In order to detect the origin of the heteroscedasticity, a second regression model with
multiple nonlinear effects was analyzed. As job control is expected to buffer the positive
effect of work pressure on emotional exhaustion (cf. Häusser, Mojzisch, Niesel, &
Schulz-Hardt, 2010; Karasek, 1979) the interaction effect of work pressure and job
control
(WP×JC)
was included in the regression equation. Additionally, quadratic
terms were included for the predictor
WP
and
JC
, because this can reduce the risk of
a spurious interaction (cf. Cortina, 1993; Klein, Schermelleh-Engel, Moosbrugger, &
Kelava, 2009).
EE =β0+β1WP+β2CR +β3JC +β4WP
2+β5JC2+β6WP×JC +e(17)
Before forming product variables, we standardized the predictor variables in order to
reduce multicollinearity and to obtain a correctly standardized solution (Aiken & West,
1991). The OLS regression for the nonlinear model yielded
R2=.51
, the standardized
regression equation is:
ˆzEE =−.07 +.26zWP +.29zCR −.29zJC +0.07zWP
2−0.06zJC2−.14zWP ×zJC.(18)
Besides significant linear effects, the quadratic effect of work pressure (with
t=2.48
,
SE =.027
,
p=.01
) and the interaction effect (with
t=−4.2
,
SE =.034
,
p<.01
)
were significant, while the quadratic effect of job control (with
t=−1.88
,
SE =.029
,
p=.06
) just failed to reach statistical significance. Compared to the linear model the
value of
hhet
was reduced to
hhet =1.14
. As this value was smaller than the critical value
(1.65) it was concluded that the residuals were now homoscedastic in the nonlinear model.
The nonlinear terms explained satisfactorily all the heteroscedasticity that appeared in
the residuals of the linear model. The
AIC
value of 996.08 also indicated an improved
model fit compared to the linear model (AIC =1019.82).
Figure 3 gives estimated histograms of
hhet
for both linear (left panel) and nonlinear
models (right panel). The
hhet
-values were estimated using bootstrapping with 10,000
replications. According to our expectations, the distribution of the resampled
hhet
-values
was shifted to the left when a nonlinear model was fit to the data. The kurtosis was -.05
for the linear model and .08 for the nonlinear model. The linear model had a skewness
of .15, the nonlinear model a skewness of .32.
Test for heteroscedasticity in regression models 587
Linear Model
−20 2 4 6
0.0 0.1 0.2 0.3 0.4
Resampled hhet
Probability Density
Nonlinear Model
−20 2 4 6
0.0 0.1 0.2 0.3 0.4
Resampled hhet
Probability Density
Figure 3: Bootstrapped hhet-values for the linear and nonlinear model of emotional
exhaustion. 10,000 data sets were resampled.
Our results of the empirical study are well in line with the Job Demands-Resources
(JD-R) model (Bakker, Demerouti, De Boer, & Schaufeli, 2003; Demerouti, Bakker,
Nachreiner, & Schaufeli, 2001), a model often used to explain how job strain (e.g.,
burnout) may be produced by two working conditions, for example, job demands and
job resources (see also Diestel & Schmidt, 2009). The results revealed a buffering effect
of job control on the relationship between work pressure and emotional exhaustion: For
high values of job control, the enhancing effect of work pressure on emotional exhaustion
is diminished. Additionally, we found a quadratic effect (
WP
2
). While this effect was
not particularly large, it indicates that the effect of quantitative work stress on burnout is
especially severe under high levels of work pressure.
Discussion
In this article, we proposed the measure
hhet
for detecting heteroscedasticity in regression
analysis. This measure utilizes the kurtosis of the residuals in a new context and makes
588 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel
direct use of the dispersion of the squared residuals. In contrast to other heteroscedas-
ticity tests (e.g., Breusch & Pagan, 1979; White, 1980), it does not require a specific
parameterization of heteroscedasticity.
In a Monte-Carlo Study we tested the performance of
hhet
. The results indicate the ability
of the measure to respond to model misspecification caused by nonlinear predictor terms
omitted in the analyzed model. A power analysis demonstrated the need of sufficiently
large sample size when small nonlinear effects are omitted. We did not investigate the
performance of
hhet
for particularly small sample sizes. It is evident from our results
that the statistical power would be too low in this case. A Type I error analysis showed
encouraging results, the Type I error rate was never higher than 5.76 % and therefore
only slightly increased. Thus, the measure
hhet
could be used in regression analysis to
identify heteroscedastic errors. For one simulation condition,
hhet
was compared to the
AIC
. The
AIC
showed the necessity of the nonlinear terms in all simulated datasets. Still,
it should be noted that the
AIC
cannot be used to detect heteroscedasticity related to
unobserved predictors. For heteroscedasticity due to omitted predictors, the power of
hhet
was considerably higher than the power of the Breusch-Pagan test. This was expected,
because the Breusch-Pagan test only detects explanatory variables that are related to
the error variances (Breusch & Pagan, 1979). On the other hand, if heteroscedasticity
is caused by the observed predictors, residual based tests such as the Breusch-Pagan
test are preferable. Still,
hhet
does also respond to this kind of heteroscedasticity, but
with lower power. The applicability of
hhet
was further demonstrated by an empirical
example from psychology, where a regression model with linear terms was shown to
have heteroscedastic error terms related to omitted nonlinear terms. The
hhet
-value
responded to the fact that the model was misspecified when nonlinear predictor terms
were omitted.
Regression models have been used in the social sciences at least since 1899, when
Yule published a paper on the causes of pauperism (Yule, 1899). At present, regression
models are state-of-the-art not only for the social and behavioral sciences, but also across
scientific disciplines. In order to enhance prediction, nonlinear effects, i.e. interaction
and quadratic effects, have been added to the linear regression equation. The use of
interaction effects has increased significantly since Aiken and West’s (1991) seminal
book on moderated regression. In psychological research overlooked or yet unidentified
moderator variables go typically along with omitted product terms in regression. Adding
an interaction term to a regression model can therefore greatly enhance the understanding
of the relationships among the variables in the model. For example, in the context of
burnout research, several studies have demonstrated buffering effects of diverse resources
on the relationship between stress and strain (cf. Gray-Stanley & Muramatsu, 2011;
Schmidt, 2007). Additionally, curvilinear effects on burnout have been found, for
example, between work ambiguity on burnout (Jamal, 2008) and between job demands
Test for heteroscedasticity in regression models 589
and anxiety (de Jonge & Schaufeli, 1998).
The present study has some important limitations. First, we examined the measure under
ideal distributional conditions where the residuals were all normally distributed. Future
simulation studies are needed to test the robustness of
hhet
to violations of the normality
assumption. Second, the effect of strong overparameterization should be investigated
in a simulation study. In practice, the researcher should pay attention to the fact that a
strongly overparameterized model can lead to wider confidence intervals.
One should keep in mind that the
hhet
measure is not constructive, which means that
a significant
hhet
-value provides no specific information about the source of the het-
eroscedasticity in the data. There may exist different possible reasons for heteroscedas-
ticity in multiple regression. One possible reason is the presence of outliers in the data,
which should be checked routinely before performing regression analysis and before
applying the measure
hhet
. In multiple regression an incorrectly specified regression
model, where important variables are omitted or where the functional form is incorrect,
may produce significant results when testing heteroscedasticity. In order to analyze this
type of heteroscedasticity, the Breusch-Pagan test is well suited when the predictors
that form the nonlinear terms are observed. The new measure
hhet
is advantageous and
could be used if nonlinear terms of unknown predictor variables are assumed to having
been omitted in the study. For this purpose, theoretical considerations about possible
model misspecifications and other potential sources of heteroscedastic residuals are
necessary.
Author note
This research was supported by Grant No. SCHE1412-1/1 from the German Research
Foundation (DFG).
References
Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting
interactions. Newbury Park, CA: Sage.
Bakker, A. B., Demerouti, E., De Boer, E., & Schaufeli, W. B. (2003). Job demands and
job resources as predictors of absence duration and frequency. Journal of Vacational
Behavior,62(2), 341–356. doi: 10.1016/S0001-8791(02)00030-1
590 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel
Breusch, T. S., & Pagan, A. R. (1979). A simple test for heteroscedasticity and random
coefficient variation. Econometrica,47(5), 1287–1294. doi: 10.2307/1911963
Büssing, A., & Perrar, K.-M. (1992). Die Messung von Burnout. Untersuchung einer
deutschen Fassung des Maslach Burnout Inventory (MBI-D)[Burnout measurement. A
Study of a German version of the Maslach Burnout Inventory (MBI-D)]. Diagnostica,
38, 328–353.
Cook, R. D., & Weisberg, S. (1982). Residuals and influence in regression. New York:
Chapman and Hall.
Cook, R. D., & Weisberg, S. (1983). Diagnostics for heteroscedasticity in regression.
Biometrika,70(1), 1–10. doi: 10.2307/2335938
Cortina, J. M. (1993). Interaction, nonlinearity and multicollinearity: Implications for
multiple regression. Journal of Management,19(4), 915 – 922.
Davidson, R., & MacKinnon, J. G. (1993). Estimation and inference in econometrics.
New York: Oxford University Press.
de Jonge, J., & Schaufeli, W. B. (1998). Job characteristics and employee well-being:
A test of Warr’s vitamin model in health care workers using structural equation
modelling. Journal of Organizational Behavior,19(4), 387–407.
Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The job
demands-resources model of burnout. Journal of Applied Psychology,86(3), 499–
512.
Diestel, S., & Schmidt, K.-H. (2009). Mediator and moderator effects of demands on
self-control in the relationship between work load and indicators of job strain. Work
& Stress,23(1), 60–79. doi: 10.1080/02678370902846686
Dijkstra, T. K., & Schermelleh-Engel, K. (2014). Consistent partial least squares
for nonlinear structural equation models. Psychometrika,79(4), 585–604. doi:
10.1007/s11336-013-9370-0
Engle, R. F. (1984). Wald, likelihood ratio and Lagrange multiplier tests in econometrics.
In Z. Griliches & M. D. Intriligator (Eds.), Handbook of Econometrics (Vol. 2, pp.
775–826). Amsterdam: Elsevier.
Gray-Stanley, J. A., & Muramatsu, N. (2011). Work stress, burnout, and social and
personal resources among direct care workers. Research in Developmental Disabilities,
32(3), 1065–1074. doi: 10.1016/j.ridd.2011.01.025
Greene, W. H. (2003). Econometric Analysis (5th ed.). Upper Saddle River, NJ: Prentice
Hall.
Test for heteroscedasticity in regression models 591
Greene, W. H. (2012). Econometric Analysis (7th ed.). Upper Saddle River, NJ: Prentice
Hall.
Häusser, J. A., Mojzisch, A., Niesel, M., & Schulz-Hardt, S. (2010). Ten years on: A re-
view of recent research on the job demand-control (-support) model and psychological
well-being. Work & Stress,24(1), 1–35. doi: 10.1080/02678371003683747
Jackson, P. R., Wall, T. D., Martin, R., & Davids, K. (1993). New measures of
job control, cognitive demand, and production responsibility. Journal of Applied
Psychology,78(5), 753–762. doi: 10.1037/0021-9010.78.5.753
Jamal, M. (2008). Burnout among employees of a multinational corporation in Malaysia
and Pakistan: An empirical examination. International Management Review,4(1),
60–71.
Karasek, R. A. (1979). Job demands, job decision latitude, and mental strain: Impli-
cations for job redesign. Administrative Science Quarterly,24(2), 285–308. doi:
10.2307/2392498
Klein, A. G., & Moosbrugger, H. (2000). Maximum likelihood estimation of latent
interaction effects with the LMS method. Psychometrika,65(4), 457–474. doi:
10.1007/BF02296338
Klein, A. G., & Muthén, B. O. (2007). Quasi-maximum likelihood estimation of
structural equation models with multiple interaction and quadratic effects. Multivariate
Behavioral Research,42(4), 647–673. doi: 10.1080/00273170701710205
Klein, A. G., & Schermelleh-Engel, K. (2010). Introduction of a new measure for
detecting poor fit due to omitted nonlinear terms in SEM. AStA Advances in Statistical
Analysis,94(2), 157-166. doi: 10.1007/s10182-010-0130-5
Klein, A. G., Schermelleh-Engel, K., Moosbrugger, H., & Kelava, A. (2009). Assessing
spurious interaction effects. In T. Teo & M. S. Khine (Eds.), Structural equation
modeling in educational research: Concepts and applications (p. 13-28). Rotterdam,
NL: Sense Publishers.
MacKinnon, J. G., & White, H. (1985). Some heteroskedasticity-consistent covariance
matrix estimators with improved finite sample properties. Journal of Econometrics,
29(3), 305-325. doi: 10.1016/0304-4076(85)90158-7
Maslach, C., & Jackson, S. E. (1981). The measurement of experienced burnout. Journal
of Occupational Behaviour,2(2), 99-113. doi: 10.1002/job.4030020205
Maslach, C., Jackson, S. E., & Leiter, M. P. (1986). Maslach Burnout Inventory (2nd
ed.). Palo Alto, CA: Consulting Psychologist Press.
592 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel
Maslach, C., & Leiter, M. P. (1997). The Truth About Burnout. San Francisco, CA:
Jossey-Bass.
Prümper, J., Hartmannsgruber, K., & Frese, M. (1995). KFZA. Kurz-Fragebogen zur
Arbeitsanalyse [Short-Questionnaire for Job Analysis]. Zeitschrift für Arbeits- und
Organisationspsychologie,39(3), 125-132.
R Core Team. (2015). R: A language and environment for statistical computing (Version
3.2.2.) [Computer software manual]. Vienna, Austria: R Foundation for Statistical
Computing.
Rosopa, P. J., Schaffer, M. M., & Schroeder, A. M. (2013). Managing heteroscedasticity
in general linear models. Psychological Methods,18(3), 335-351. doi: 10.1037/
a0032553
Schmidt, K.-H. (2004). Formen der Kontrolle als Puffer der Belastungs-Beanspruchungs-
Beziehung [Forms of control as buffer of the relationship between job demands and
strain]. Zeitschrift für Arbeitswissenschaft,58, 44-52.
Schmidt, K.-H. (2007). Organizational commitment: A further moderator in the rela-
tionship between work-stress and strain? International Journal of Stress Management,
14(4), 26–40. doi: 10.1037/1072-5245.14.1.26
Schmidt, K.-H., & Neubach, B. (2009). Selbstkontrollanforderungen als spezifische
Belastungsquelle bei der Arbeit [Self-control demands as a specific source of stress at
work]. Zeitschrift für Personalpsychologie,8(4), 169-179. doi: 10.1026/1617-6391.8
.4.169
Stevens, J. P. (1984). Outliers and influential data points in regression analysis. Psycho-
logical Bulletin,95(2), 334-344. doi: 10.1037/0033-2909.95.2.334
Wald, A. (1943). Tests of statistical hypotheses concerning several parameters when the
number of observations is large. Transactions of the American Mathematical Society,
54(3), 426-482.
White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and
a direct test for heteroskedasticity. Econometrica,48(4), 817–838. doi: 10.2307/
1912934
Yule, G. U. (1899). An investigation into the causes of changes in pauperism in England,
chiefly during the last two intercensal decades (part I.). Journal of the Royal Statistical
Society,62(2), 249–295. doi: 10.2307/2979889